Typically, such Morita situations involve three ingredients: a ‘syntactic’ ground level to which the respective concept of Morita equivalence applies, a ‘hypersyntactic’ level that obtains from an ‘idempotent’ completion, and a second process of completion to a ‘semantic’ level where the equivalence relation for the syntactic ground level is defined by plain equivalence of category e.g. Morita equivalence for small categories is defined as equivalence of their presheaf categories with Cauchy completion as intermediate hypersyntactic level.

So the broad intuition is that Morita equivalence is a coarse grained semantic equivalence that obtains between syntactic gadgets - basically two theories that have up to equivalence the same category of models. The role of the intermediate hypersyntactic level in this analogy is that of an ‘ideal syntax’ (syntax classifier) that already reflects the relations at the semantic level. The categorical equivalence (via bimodules) from the semantic level then shows up at the intermediate level as a (‘Cauchy convergent’∼\sim ‘fgp-module’) bidirectional translation from one syntax into another.

Classical Morita theorem

Given rings RR and SS, the following properties are equivalent

The categories of left SS-modules and left RR-modules are equivalent;

The categories of right SS-modules and right RR-modules are equivalent;

There are bimodules RMS{}_R M_S and SNR{}_S N_R such that ⊗RM\otimes_R M and ⊗SN\otimes_S N form an adjoint equivalence between the category of right SS- and the category of right RR-modules;

The ring RR is isomorphic to the endomorphism ring of a generator in the category of left (or right) SS-modules;

The ring SS is isomorphic to the endomorphism ring of a generator in the category of left (or right) RR-modules.

An important weakening of the Morita equivalence is Morita context (in older literature sometimes called pre-equivalence).

Definitions

In algebra

Two rings are Morita equivalent if the equivalent statements in the Morita theorem above are true. A Morita equivalence is a weakly invertible 1-cell in the bicategory Rng\mathrm{Rng} of rings, bimodules and morphisms of bimodules.

A theorem in ring theory says that the center of a ring is isomorphic to the center of its category of modules and that Morita equivalent rings have isomorphic centers. Especially, two commutative rings are Morita equivalent precisely when they are isomorphic!

This shows that the property of having center ZZ up to isomorphism is stable within Morita equivalence classes. Properties of this kind are sufficiently important to deserve a special name:

A property PP of rings is called a Morita invariant iff whenever PP holds for a ring RR, and RR and SS are Morita equivalent then PP also holds for SS. Another classical example is the property of being simple. (cf. Cohn 2003)

In homotopy theory

In any homotopy theory framework a Morita equivalence between objects CC and DD is a span

In Lie groupoid theory

Lie groupoids up to Morita equivalence are equivalent to differentiable stacks. This relation between Lie groupoids and their stacks of torsors is analogous to the relation between algebras and their categories of modules, which is probably the reason for the choice of terminology.